Oscillation and nonoscillation of perturbed nonlinear equations with p-Laplacian

In this paper, we analyze oscillatory properties of perturbed half-linear differential equations (i.e., equations with one-dimensional p-Laplacian). The presented research covers the Euler and Riemann–Weber type equations with very general coefficients. We prove an oscillatory result and a nonoscillatory one, which show that the studied equations are conditionally oscillatory (i.e., there exists a certain threshold value that separates oscillatory and nonoscillatory equations). The obtained criteria are easy to use. Since the number of perturbations is arbitrary, we solve the oscillation behavior of the equations in the critical setting when the coefficients give exactly the threshold value. The results are new for linear equations as well.